## G(x) = min(f(t)) and g(x)=max(f(t))

If

$f_{1}(x) = \left\{\begin{matrix} min \left \{ x^2, \left | x \right | \right \} & for \left | x \right |\leq 1\\ max \left \{ x^2, \left | x \right | \right \} & for \left | x \right |\ > 1 \end{matrix}\right.$

$f_{2}(x) = \left\{\begin{matrix} min \left \{ x^2, \left | x \right | \right \} & for \left | x \right |\ > 1\\ max \left \{ x^2, \left | x \right | \right \} & for \left | x \right |\leq 1 \end{matrix}\right.$

and $f(x)=f_{1}(x)-2f_{2}(x)$

Then draw the graph of

$g(x)=\left\{\begin{matrix} min\left \{ f(t) : -3\leqslant t\leq x, -3\leqslant x< 0 \right \}\\ max\left \{ f(t) : 0\leqslant t\leq x,0\leqslant x\leqslant 3\right \} \end{matrix}\right.$

So lets draw the graph of $latex $f_{1}(x)$,$latex $f_{2}(x)$ and $g(x)$

The graph of f(x)

Now let’s draw the graph of g(x)

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